# Convergence of implicit schemes for Hamilton-Jacobi-Bellman quasi-variational inequalities

## TL;DR

This paper provides rigorous convergence proofs for implicit numerical schemes solving Hamilton-Jacobi-Bellman quasi-variational inequalities, using a novel nonlocal consistency approach and Barles-Souganidis analysis.

## Contribution

It introduces a new nonlocal consistency framework and proves convergence of implicit schemes for HJBQVIs, filling a gap in existing computational methods.

## Key findings

- Established convergence of implicit schemes for HJBQVIs
- Introduced the concept of nonlocal consistency
- Proved results using comparison principles

## Abstract

In [Azimzadeh, P., and P. A. Forsyth. "Weakly chained matrices, policy iteration, and impulse control." SIAM J. Num. Anal. 54.3 (2016): 1341-1364], we outlined the theory and implementation of computational methods for implicit schemes for Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVIs). No convergence proofs were given therein. This work closes the gap by giving rigorous proofs of convergence. We do so by introducing the notion of nonlocal consistency and appealing to a Barles-Souganidis type analysis. Our results rely only on a well-known comparison principle and are independent of the specific form of the intervention operator.